Number of Questions: 10
Each question is worth 10 marks.
Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) information previously stored by students (such as formulas, programs, notes, etc.), (iv) a computer algebra system, (v) dynamic geometry software.
Parts of each question can be of two types:
SHORT ANSWER parts indicated by
worth 3 marks each
full marks are given for a correct answer which is placed in the box
part marks are awarded if relevant work is shown in the space provided
FULL SOLUTION parts indicated by
worth the remainder of the 10 marks for the question
must be written in the appropriate location in the answer booklet
marks awarded for completeness, clarity, and style of presentation
a correct solution poorly presented will not earn full marks
WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.
Extra paper for your finished solutions supplied by your supervising teacher must be
inserted into your answer booklet. Write your name, school name, and question number
on any inserted pages.
Express answers as simplified exact numbers except where otherwise indicated. For example, and are simplified exact numbers.
Do not discuss the problems or solutions from this contest online for the next 48 hours.The name, grade, school and location, and score range of some top-scoring students will be
published on our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location,
and score of some top-scoring students may be shared with other mathematical organizations
for other recognition opportunities.
NOTE:
Please read the instructions for the contest.
Write all answers in the answer booklet provided.
For questions marked , place your answer in the appropriate box in the answer booklet and show your work.
For questions marked , provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution.
Diagrams are not drawn to scale. They are intended as aids only.
While calculators may be used for numerical calculations, other mathematical steps must
be shown and justified in your written solutions, and specific marks may be allocated for
these steps. For example, while your calculator might be able to find the -intercepts of the graph of an equation like , you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down.
Joyce has two identical jars. The first jar is full of water and contains 300 mL of water. The second jar is full of water. How much water, in mL, does the second jar contain?
What integer satisfies ?
If , determine all possible values of .
In the diagram, two small circles of radius 1 are tangent to each other and to a larger circle of radius 2.
What is the area of the shaded region?
Kari jogs at a constant speed of 8 km/h. Mo jogs at a constant speed of 6 km/h. Kari and Mo jog from the same starting point to the same finishing point along a straight road. Mo starts at 10:00 a.m. Kari and Mo both finish at 11:00 a.m. At what time did Kari start to jog?
The line with equation is parallel to the line with equation . The line with equation passes through the point . Determine the value of .
Michelle calculates the average of the following numbers: Daphne removes one number and calculates the average of the remaining numbers. The average that Daphne calculates is one less than the average that Michelle calculates. Which number does Daphne remove?
If , what is the value of ?
Suppose that . Determine the value of .
In the diagram, has a right angle at and point lies on .
If , and , what is the length of ?
The points and both lie on a circle centered at the origin. Determine the possible values of .
Determine the two pairs of positive integers with that satisfy the equation .
Consider the system of equations: Determine the number of integers with for which there is at least one pair of integers that is a solution to the system.
A regular pentagon covers part of another regular polygon, as shown.
Visualize the pentagon on top of the polygon. Now slide the pentagon off the polygon until part of the polygon is visible.
The polygon can be seen jutting out behind two attached sides of the pentagon.
Denote the three vertices on these two attached polygon sides as P, Q, and R. The vertex Q of the pentagon is inside the pentagon.
The vertex Q of the pentagon is inside the polygon.
The first and fifth visible sides of the polygon intersect the pentagon close to the vertices P and R.
The angle from the first visible side of the polygon to the closest pentagon side is denoted a degrees.
Similarly, the angle from the fifth visible side of the polygon to the closest pentagon side is denoted b degrees.
This regular polygon has sides, five of which are completely or partially visible. In the diagram, the sum of the measures of the angles marked and is . Determine the value of .
(The side lengths of a regular polygon are all equal, as are the measures of its interior angles.)
In trapezoid , is parallel to and is perpendicular to .
Also, the lengths of , and , in that order, form a geometric sequence. Prove that is perpendicular to .
(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant.)
Determine all real numbers for which .
Consider the function . Determine all real numbers that satisfy the equation .
A circle has centre and radius 1. Quadrilateral has all 4 sides tangent to the circle at points , , , and , as shown.
Also, . If , determine the length of .
Suppose that satisfies and . Determine all possible values of , expressing your answers in the form where are integers.
For positive integers and , define . For example, the value of is .
Determine the value of .
Determine all positive integers for which is an integer.
If and are positive integers and is an integer, prove that must be a multiple of 3.
Determine four pairs of positive integers , with , for which is an integer.
Amir and Brigitte play a card game. Amir starts with a hand of 6 cards: 2 red, 2 yellow and 2 green. Brigitte starts with a hand of 4 cards: 2 purple and 2 white. Amir plays first. Amir and Brigitte alternate turns. On each turn, the current player chooses one of their own cards at random and places it on the table. The cards remain on the table for the rest of the game. A player wins and the game ends when they have placed two cards of the same colour on the table. Determine the probability that Amir wins the game.
Carlos has coins, numbered to . Each coin has exactly one face called “heads”. When flipped, coins land heads with probabilities , respectively. When Carlos flips each of the coins exactly once, the probability that an even number of coins land heads is exactly . Must there be a between 1 and 14, inclusive, for which ? Prove your answer.
Serge and Lis each have a machine that prints a digit from 1 to 6. Serge’s machine prints the digits with probability , , , , , , respectively. Lis’s machine prints the digits with probability , , , , , , respectively. Each of the machines prints one digit. Let be the probability that the sum of the two digits printed is . If and , prove that and .
Further Information
For students...
Thank you for writing the Euclid Contest!
If you are graduating from secondary school, good luck in your future endeavours! If you will be returning to secondary school next year, encourage your teacher to register you for the Canadian Senior Mathematics Contest, which will be written in November.